TITLE: What is the probability that a number chosen at random in $[0,1]$ is transcendental?
QUESTION [6 upvotes]: Consider the interval $[0,1]$. What is the probability that a number chosen at random in $[0,1]$ is transcendental?
Please give me some points on how to start this problem.

REPLY [7 votes]: I realize this question is old but I wanted to give my explanation:

The measure of the unit interval $[0,1]$ is $1$.
Cantor proved that the algebraic numbers are countable, and thus the measure of the set of algebraics on $[0,1]$ is the countable union of the measures of the singletons. Since each singleton has measure zero, the set of algebraic numbers on $[0,1]$ has measure zero.
Since the algebraic numbers and the transcendental numbers both partition $[0,1]$, and the set of algebraic numbers has measure zero, the set of transcendental numbers on $[0,1]$ must be of measure $1$.

So the probability of picking a transcendental number on the unit interval is $1$. Measure of a subset of $[0,1]$ more or less translates to "probability".